I am not surprised by the term since I'm familiar with the Greek Catholic Church of Western Ukraine and its rather recent problems.
As to Bratislava, there exists indeed the Byzantine Catholic Parish of Bratislava, Church of the Exaltation of the Venerable Cross.
Maybe this can help:
"At the time of the early Church, there were several rich cultures in the Middle East and each of them has given rise to a different church tradition. The traditions of this church reflect the Greek or Byzantine culture, and so we are called Greek Catholics or Byzantine Catholics (from Byzantium, the ancient name for Constantinople).
Greek Catholics in the Middle East were also nicknamed 'Melkite' because they followed the faith of the Byzantine emperor, or melek. In addition, many Greek Catholic Churches are identified by their national origin, such as Hungarian, Romanian or Ukrainian. "
http://www.byzantines.net